Mutually orthogonal Latin squares

Two latin squares are said to be orthogonal if no pair of corresponding elements occurs more than once. For example,

1, 2, 3
2, 3, 1
3, 1, 2

is orthogonal to

1, 3, 2
2, 1, 3
3, 2, 1

We can see this most easily by writing them together as follows, and observing that no pair appears twice:

11, 23, 32
22, 31, 13
33, 12, 21

A set of n latin squares is mutually orthogonal if every pair of latin squares from the set is orthogonal.

Euler studied orthogonal latin squares because they can be used to construct magic squares, and he found that, while orthogonal latin squares of any odd order are easy to generate, even orders are not. He conjectured, but did not prove, that there are no orthogonal latin squares of order 4n+2, for any integer n. In 1960, this was proved incorrect; it turns out that there are orthogonal latin squares of any size except 1, 2, and, oddly enough, 6.